Many, many mouthpieces have tables that aren’t flat.  This can cause major playability problems.  It may be due to manufacturing defects, warping (hard rubber mouthpieces), corrosion (metal mouthpieces), or other damage.  Also, some mouthpieces are intentionally made with “concave” tables.

This is a mouthpiece after one pass across sandpaper on a glass plate.  The lighter color is where the table contacted the sandpaper, the darker area is a big low spot where the table wouldn’t be making contact with the reed.

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A few more passes showed the next-highest spot:

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Getting better, but there’s still a low spot at the “U” of the window, where air will probably just leak out and make it hard to play:

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That last section didn’t flatten evenly, so even though it’s mostly flat, it’s really uneven where the table will meet the facing curve.

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A little more work, and the table is flat.  The transition from the flat section to the facing curve (on the side rails) is pretty even at this point, so I could start measuring and making a plan for the facing.